Statistical comparison of shear wave speed recovery using the direct algorithm and the arrival time algorithm

Abstract

Elastography is a non-invasive method which images the mechanical properties of tissue, such as tissue stiffness. Often this is accomplished by generating a shear wave that propagates in the tissue. Using the displacement of the propagating wave, a movie is created from sequences of ultrasound RF/IQ data sets. Shear wave speed is imaged utilizing this movie. Here we consider two methods used to recover the shear wave speed using the arrival time of the wave (calculated from the displacements). In particular, we consider the effect of noise in the arrival times on the recovered shear wave speed.

The Arrival Times Algorithm is derived from the Eikonal equation, but uses distance between contours of the arrival time data, coupled with the fundamental relationship "speed = distance / time," to recover shear wave speed. The resulting pdf of the recovered shear wave speed appears nearly Gaussian, with a jump discontinuity resulting from the grid discretization.

For simplicity, we reduce the problem to a plane wave, traveling at constant speed (later extended to vertically layered media). We assume Gaussian noise in the arrival times. A bound on the grid spacing is found which restricts the noisy arrival times to a column surrounding their original location. This bound will depend on the noise variance and wave speed, as well as the size of the medium. The distribution of the recovered shear wave speed is found using both the Eikonal equation and the Arrival Time Algorithm.

The Direct Algorithm utilizes the Eikonal equation, where the shear wave speed is inversely proportional to the magnitude of the gradient of the arrival times, and discretizes the derivatives using centered differences. The pdf of the recovered shear wave speed in this case has finite mean, but infinite variance.